Membrane filters provide immediate solutions to many urgent problems such as water purification, and effective remedies to pressing environmental concerns such as waste and air treatment. The ubiquity of applications gives rise to a significant amount of research in membrane material selection and structural design to optimize filter efficiency. As physical experiments tend to be costly, numerical simulation and analysis of fluid flow, foulant transport and geometric evolution due to foulant deposition in complex geometries become particularly relevant. In this dissertation, several mathematical modeling and analytical aspects of the industrial membrane filtration process are investigated. A first-principles mathematical model for fluid flow and contaminant advection/deposition through a network of cylindrical pores, and time evolution of membrane pore geometry, is proposed, formulated as a system of ordinary and partial differential equations. Membrane filter performance metrics, including total throughput (total volume of filtered fluid) and foulant concentration at membrane pore outlets, among others, are thoroughly studied against membrane geometric features such as porosity and tortuosity (average normalized distance traveled by fluid through pores between membrane top and bottom surfaces). The influence of the underlying, often complex, pore geometries on the performance of the membrane filters is explored in the following setups: (1) layered planar membrane structures with intra-layer pore connections; (2) general pore networks generated by a random graph generation protocol; (3) pore size variations in a pore network and (4) pore size gradient in a banded membrane network. Future work should include studying pore size variations on porosity graded networks and stochastic modeling of large-particle sieving in pore networks.
In Chapter 1, an overview of the experimental, computational and theoretical literature on membrane filtration is given to motivate the following Chapters. In Chapter 2, a mathematical model is proposed for multilayered membrane filters with interconnected pores in the junction between layers. A side-by-side comparison is carried out between three simple geometries that have various degrees of pore connectivity and the same initial pore radius in each layer. Pore size heterogeneities, modeled as a random perturbation on initial pore size, are also studied in detail. Via variations in the strength of the pore-size perturbation, the statistical and physical influence on key properties of membrane filters, such as initial resistance, total throughput and foulant concentration at pore outlets, are analyzed and discussed. This work appeared in Journal of Fluid Mechanics.
In Chapters 3 and 4, a random graph generation protocol is devised to generate pore networks that generalize the structures considered in Chapter 2. A membrane filter is modeled as a graph with vertices and edges representing pore junctions and pore throats respectively. Local fluid and foulant transport equations are posed on each edge, coupled with conservation laws to produce global equations that capture the connectivity of the network. When a uniform initial pore radius is assumed (Chapter 3), initial membrane porosity is found to be a strong predictor for total throughput via a power law; and accumulated foulant concentration at membrane pore outlets satisfies a negative exponential relationship to membrane tortuosity. When pore size variations are imposed as pore-wise noise perturbation, however (Chapter 4), it is observed that network variations induced from the random graph generation have a stronger influence on membrane performance, unless noise strength is large. Membrane initial porosity is again found to be a crucial geometric feature. The work of these Chapters appeared in SIAM Journal on Applied Mathematics and Journal of Membrane Science, respectively.
In Chapter 5, a variant of the protocol described in Chapter 3 is developed to generate banded pore networks in which the pore radius decreases from one band to the next, creating a pore-size gradient. Under specific assumptions, an optimal radius gradient in the depth of the banded membrane that maximizes either total throughput of filtrate or the particle retention capability of the membrane, is found. Finally, in Chapter 6, conclusions from the previous chapters are discussed, along with two open questions for future work.
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